It is best to avoid the term ‘monad’ for this concept on this wiki, since it has more or less nothing to with the categorial monads that are all over the place here (including elsewhere on this very page).

For ringed spaces

Consider a morphism (f,f♯):(Y,𝒪Y)→(X,𝒪X)(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X) of ringed spaces for which the corresponding map f♯:f*𝒪X→𝒪Yf^\sharp:f^*\mathcal{O}_X\to\mathcal{O}_Y of sheaves on YY is surjective. Let ℐ=ℐf=Kerf♯\mathcal{I} = \mathcal{I}_f = Ker\,f^\sharp, then 𝒪Y=f♯(𝒪X)/ℐf\mathcal{O}_Y = f^\sharp(\mathcal{O}_X)/\mathcal{I}_f. The ring f*(𝒪Y)f^*(\mathcal{O}_Y) has the ℐ\mathcal{I}-preadic filtration which has the associated graded ring Gr•=⊕nℐfn/ℐfn+1Gr_\bullet =\oplus_{n} \mathcal{I}_f^n/\mathcal{I}^{n+1}_f which in degree 11 gives the conormal sheafGr1=ℐf/ℐf2Gr_1 = \mathcal{I}_f/\mathcal{I}^2_f of ff. The 𝒪Y\mathcal{O}_Y-augmented ringed space (Y,f♯(𝒪X)/ℐn+1)(Y,f^\sharp(\mathcal{O}_X)/\mathcal{I}^{n+1}) is called the nn-th infinitesimal neighborhood of YY along morphism ff. Its structure sheaf is called the nn-th normal invariant of ff.